Do the two things simultaneously. The culture is embedded in the language. So mastery of the language provides a key to this culture, while at the same time, understanding of the culture will.facilitate your language-learning. Language-learning is not only about putting words together, it is about acquiring a way of life and a way of thought.

when we learn a language, I think we do should learn its culture, but it does not matter you learn it before or at the same time when you are learning the language

Indeed, it is acquired test if people are different, there is no one can answer this question as well as you experience. Some like to learn from you, but some just want you to learn from them or others for other motives. So how can I answer?